Computes the pdf, cdf, quantile, and random numbers, draws the q-q plot, and estimates the parameters of the Marshall-Olkin Kumaraswamy G distribution. General form for the probability density function (pdf) of the Marshall-Olkin Kumaraswamy G distribution due to Roshini and Thobias (2017) is given by
$$f(x,{\Theta}) = \frac{{abd g(x-\mu,\theta ){{\left( {G(x-\mu,\theta )} \right)}^{a - 1}}{{\left[ {1 - {{\left( {G(x-\mu,\theta )} \right)}^a}} \right]}^{b - 1}}}}{{{{\left[ {1 - \left( {1 - d} \right){{\left[ {1 - {{\left( {G(x-\mu,\theta )} \right)}^a}} \right]}^b}} \right]}^2}}},$$
where \(\theta\) is the baseline family parameter vector. Also, a>0, b>0, d>0, and \(\mu\) are the extra parameters induced to the baseline cumulative distribution function (cdf) G whose pdf is g. The general form for the cumulative distribution function (cdf) of the Marshall-Olkin Kumaraswamy G distribution is given by
$$F(x,{\Theta}) = 1 - \frac{{d{{\left[ {1 - {{\left( {G(x-\mu,\theta )} \right)}^a}} \right]}^b}}}{{1 - \left( {1 - d} \right){{\left[ {1 - {{\left( {G(x-\mu,\theta )} \right)}^a}} \right]}^b}}}.$$
Here, the baseline G refers to the cdf of famous families such as: Birnbaum-Saunders, Burr type XII, Exponential, Chen, Chisquare, F, Frechet, Gamma, Gompertz, Linear failure rate (lfr), Log-normal, Log-logistic, Lomax, Rayleigh, and Weibull. The parameter vector is \(\Theta=(a,b,d,\theta,\mu)\) where \(\theta\) is the baseline G family's parameter space. If \(\theta\) consists of the shape and scale parameters, the last component of \(\theta\) is the scale parameter (here, a, b, and d are the first, second, and the third shape parameters). Always, the location parameter \(\mu\) is placed in the last component of \(\Theta\).
dmokumg(mydata, g, param, location = TRUE, log=FALSE)
pmokumg(mydata, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
qmokumg(p, g, param, location = TRUE, log.p = FALSE, lower.tail = TRUE)
rmokumg(n, g, param, location = TRUE)
qqmokumg(mydata, g, location = TRUE, method)
mpsmokumg(mydata, g, location = TRUE, method, sig.level)The name of family's pdf including: "birnbaum-saunders", "burrxii", "chisq", "chen", "exp", "f", "frechet", "gamma", "gompetrz", "lfr", "log-normal", "log-logistic", "lomax", "rayleigh", and "weibull".
a vector of value(s) between 0 and 1 at which the quantile needs to be computed.
number of realizations to be generated.
Vector of observations.
parameter vector \(\Theta=(a,b,d,\theta,\mu)\)
If FALSE, then the location parameter will be omitted.
If TRUE, then log(pdf) is returned.
If TRUE, then log(cdf) is returned and quantile is computed for exp(-p).
If FALSE, then 1-cdf is returned and quantile is computed for 1-p.
The used method for maximizing the sum of log-spacing function. It will be "BFGS", "CG", "L-BFGS-B", "Nelder-Mead", or "SANN".
Significance level for the Chi-square goodness-of-fit test.
A vector of the same length as mydata, giving the pdf values computed at mydata.
A vector of the same length as mydata, giving the cdf values computed at mydata.
A vector of the same length as p, giving the quantile values computed at p.
A vector of the same length as n, giving the random numbers realizations.
A sequence of goodness-of-fit statistics such as: Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian Information Criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramer-von Misses statistic (CM), Anderson Darling statistic (AD), log-likelihood statistic (log), and Moran's statistic (M). The Kolmogorov-Smirnov (KS) test statistic and corresponding p-value. The Chi-square test statistic, critical upper tail Chi-square distribution, related p-value, and the convergence status.
It can be shown that the Moran's statistic follows a normal distribution. Also, a chi-square approximation exists for small samples whose mean and variance approximately are m(log(m)+0.57722)-0.5-1/(12m) and m(\(\pi^2\)/6-1)-0.5-1/(6m), respectively, with m=n+1, see Cheng and Stephens (1989). So, a hypothesis tesing can be constructed based on a sample of n independent realizations at the given significance level, indicated in above as sig.level.
Cheng, R. C. H. and Stephens, M. A. (1989). A goodness-of-fit test using Moran's statistic with estimated parameters, Biometrika, 76 (2), 385-392.
Roshini, G. and Thobias, S. (2017). Marshall-Olkin Kumaraswamy Distribution, International Mathematical Forum, 12 (2), 47-69.
# NOT RUN {
mydata<-rweibull(100,shape=2,scale=2)+3
dmokumg(mydata, "weibull", c(1,1,1,2,2,3))
pmokumg(mydata, "weibull", c(1,1,1,2,2,3))
qmokumg(runif(100), "weibull", c(1,1,1,2,2,3))
rmokumg(100, "weibull", c(1,1,1,2,2,3))
qqmokumg(mydata, "weibull", TRUE, "Nelder-Mead")
mpsmokumg(mydata, "weibull", TRUE, "Nelder-Mead", 0.05)
# }
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